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Properties

Label 20.0.15705390275...3769.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 11^{18}$
Root discriminant $22.90$
Ramified primes $7, 11$
Class number $5$
Class group $[5]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![947, -1645, 3430, -5370, 8019, -9087, 8630, -7225, 5062, -2046, -714, 2145, -1963, 965, -89, -276, 264, -138, 47, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 264*x^16 - 276*x^15 - 89*x^14 + 965*x^13 - 1963*x^12 + 2145*x^11 - 714*x^10 - 2046*x^9 + 5062*x^8 - 7225*x^7 + 8630*x^6 - 9087*x^5 + 8019*x^4 - 5370*x^3 + 3430*x^2 - 1645*x + 947)
 
gp: K = bnfinit(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 264*x^16 - 276*x^15 - 89*x^14 + 965*x^13 - 1963*x^12 + 2145*x^11 - 714*x^10 - 2046*x^9 + 5062*x^8 - 7225*x^7 + 8630*x^6 - 9087*x^5 + 8019*x^4 - 5370*x^3 + 3430*x^2 - 1645*x + 947, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 47 x^{18} - 138 x^{17} + 264 x^{16} - 276 x^{15} - 89 x^{14} + 965 x^{13} - 1963 x^{12} + 2145 x^{11} - 714 x^{10} - 2046 x^{9} + 5062 x^{8} - 7225 x^{7} + 8630 x^{6} - 9087 x^{5} + 8019 x^{4} - 5370 x^{3} + 3430 x^{2} - 1645 x + 947 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1570539027548129147161113769=7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.90$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{45} a^{12} - \frac{2}{15} a^{11} - \frac{7}{45} a^{10} - \frac{2}{15} a^{7} + \frac{7}{45} a^{6} - \frac{2}{15} a^{5} + \frac{1}{3} a^{4} - \frac{22}{45} a^{2} - \frac{2}{15} a - \frac{2}{45}$, $\frac{1}{45} a^{13} + \frac{2}{45} a^{11} + \frac{1}{15} a^{10} - \frac{2}{15} a^{8} + \frac{1}{45} a^{7} + \frac{2}{15} a^{6} + \frac{1}{5} a^{5} + \frac{1}{3} a^{4} + \frac{8}{45} a^{3} + \frac{4}{15} a^{2} - \frac{8}{45} a + \frac{1}{15}$, $\frac{1}{765} a^{14} - \frac{7}{765} a^{13} - \frac{1}{765} a^{12} + \frac{97}{765} a^{11} - \frac{2}{17} a^{10} - \frac{32}{255} a^{9} - \frac{92}{765} a^{8} + \frac{2}{765} a^{7} + \frac{22}{255} a^{6} + \frac{2}{17} a^{5} + \frac{248}{765} a^{4} - \frac{149}{765} a^{3} - \frac{131}{765} a^{2} + \frac{62}{765} a + \frac{22}{51}$, $\frac{1}{765} a^{15} + \frac{1}{765} a^{13} + \frac{1}{153} a^{12} - \frac{74}{765} a^{11} + \frac{22}{765} a^{10} + \frac{1}{765} a^{9} + \frac{8}{85} a^{8} - \frac{124}{765} a^{7} + \frac{8}{765} a^{6} + \frac{317}{765} a^{5} + \frac{19}{255} a^{4} - \frac{256}{765} a^{3} - \frac{158}{765} a^{2} + \frac{101}{765} a - \frac{172}{765}$, $\frac{1}{3825} a^{16} + \frac{2}{3825} a^{15} - \frac{2}{3825} a^{14} - \frac{2}{1275} a^{13} + \frac{8}{1275} a^{12} + \frac{107}{765} a^{11} + \frac{128}{3825} a^{10} - \frac{148}{3825} a^{9} - \frac{104}{765} a^{8} + \frac{97}{765} a^{7} + \frac{16}{3825} a^{6} + \frac{23}{765} a^{5} - \frac{1396}{3825} a^{4} + \frac{52}{255} a^{3} - \frac{4}{255} a^{2} + \frac{1391}{3825} a - \frac{1096}{3825}$, $\frac{1}{3825} a^{17} - \frac{1}{3825} a^{15} - \frac{2}{3825} a^{14} + \frac{41}{3825} a^{13} + \frac{2}{3825} a^{12} + \frac{473}{3825} a^{11} - \frac{61}{425} a^{10} - \frac{73}{1275} a^{9} + \frac{122}{765} a^{8} + \frac{211}{3825} a^{7} + \frac{42}{425} a^{6} + \frac{1744}{3825} a^{5} + \frac{32}{3825} a^{4} + \frac{37}{153} a^{3} + \frac{466}{3825} a^{2} - \frac{1588}{3825} a - \frac{22}{425}$, $\frac{1}{11475} a^{18} - \frac{2}{3825} a^{14} - \frac{38}{3825} a^{13} - \frac{53}{11475} a^{12} - \frac{553}{3825} a^{11} + \frac{158}{3825} a^{10} - \frac{59}{425} a^{9} + \frac{2}{3825} a^{8} + \frac{31}{3825} a^{7} - \frac{62}{459} a^{6} - \frac{37}{1275} a^{5} - \frac{584}{1275} a^{4} + \frac{242}{3825} a^{3} - \frac{596}{3825} a^{2} - \frac{1184}{3825} a + \frac{32}{675}$, $\frac{1}{293082975} a^{19} + \frac{12761}{293082975} a^{18} - \frac{10628}{97694325} a^{17} - \frac{1072}{19538865} a^{16} - \frac{53674}{97694325} a^{15} - \frac{26929}{97694325} a^{14} - \frac{2171}{293082975} a^{13} + \frac{309022}{58616595} a^{12} + \frac{11418281}{97694325} a^{11} - \frac{4034036}{97694325} a^{10} + \frac{5874913}{97694325} a^{9} + \frac{344363}{97694325} a^{8} + \frac{64487}{17240175} a^{7} - \frac{6402218}{58616595} a^{6} + \frac{7949512}{19538865} a^{5} - \frac{11953237}{32564775} a^{4} + \frac{30945521}{97694325} a^{3} - \frac{47704693}{97694325} a^{2} - \frac{134039666}{293082975} a - \frac{104209174}{293082975}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}$, which has order $5$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 111683.107049 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-7}, \sqrt{-11})\), 5.1.717409.1 x5, 10.0.3602729712967.3, 10.2.39630026842637.1 x5, 10.0.5661432406091.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.10.9.10$x^{10} + 24057$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.10$x^{10} + 24057$$10$$1$$9$$C_{10}$$[\ ]_{10}$